Higher Order Derivatives of Laplace Transform/Examples/Example 1

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Examples of Use of Laplace Transform of Integral

Let $\laptrans f$ denote the Laplace transform of the real function $f$.


$\laptrans {t^2 e^{2 t} } = \dfrac 2 {\paren {s - 2}^3}$


Proof

\(\displaystyle \paren {-1}^2 \laptrans {t^2 e^{2 t} }\) \(=\) \(\displaystyle \dfrac {\d^2} {\d s^2} \laptrans {e^{2 t} }\) Laplace Transform of Integral
\(\displaystyle \leadsto \ \ \) \(\displaystyle \laptrans {t^2 e^{2 t} }\) \(=\) \(\displaystyle \dfrac {\d^2} {\d s^2} \dfrac 1 {s - 2}\) Laplace Transform of Exponential
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 2 {\paren {s - 2}^3}\) simplification

$\blacksquare$


Sources